4. (a) Find a linear transformation with the given properties and compute $T\begin{bmatrix} a & b \\ c & d \end{bmatrix}$.
$T: M_{22} \rightarrow \mathbb{R}$
$T\begin{bmatrix} 1 & 0 \\ 0 & 0 \end{bmatrix} = 3, T\begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix} = -1, T\begin{bmatrix} 1 & 0 \\ 1 & 0 \end{bmatrix} = 0, T\begin{bmatrix} 0 & 0 \\ 0 & 1 \end{bmatrix} = 0$,
You may assume that $B = \left\{ \begin{bmatrix} 1 & 0 \\ 0 & 0 \end{bmatrix}, \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix}, \begin{bmatrix} 1 & 0 \\ 1 & 0 \end{bmatrix}, \begin{bmatrix} 0 & 0 \\ 0 & 1 \end{bmatrix} \right\}$ is a basis for
$M_{22}$ however, it is expected you are able to prove that on your own!
